EMTH118
FORMULA SHEET
TRIGONOMETRY
sin (A ± B) = sin A cos B ± cos A sin B
sin C + sin D = 2 sin
C + D cos
C − D
cos (A ± B) = cos A cos B ∓ sin A sin B
sin C − sin D = 2 cos
C + D
sin
C − D
2 sin A cos B = sin (A + B) + sin (A − B)
cos C + cos D = 2 cos
2
2
C + D cos
2
C + D
2 cos A cos B = cos (A + B) + cos (A − B)
cos C − cos D = 2 sin
2 sin A sin B = cos (A − B) − cos (A + B)
cos 2x = 2 cos2 x − 1 = 1 − 2 sin2 x
sin2 x + cos2 x = 1
sin 2x = 2 sin x cos x
tan2 x + 1 = sec2 x
cot2 x + 1 = csc2 x
DERIVATIVES
2
sin
2
2
C − D 2
D − C 2
INTEGRALS
R
f (x)
f ′ (x)
f (x)
xn
nxn−1
xn
xn+1
n+1
ln x
1/x
1/x
ln |x| + C
ex
ex
√
sin x
cos x
1
1 + x2
cos x
− sin x
Integration by parts
tan x
sec2 x
1
1 − x2
f (x) dx
sin−1 (x) + C
tan−1 (x) + C
Z
Z
u dv = uv − v du
DIFFERENTIATION RULES
csc x
− csc x cot x
(u v)′ = u′ v + uv ′
sec x
sec x tan x
u ′
cot x
− csc2 x
v
=
u′ v − uv ′
v2
QUADRATIC
FORMULA
x=
VECTORS
x · y = x1 y1 + x2 y2 + . . . + xn yn
kxk =
p
x21 + x22 + . . . + x2n
x × y = (x2 y3 − x3 y2 , −(x1 y3 − x3 y1 ), x1 y2 − x2 y1 )
x·y
kxk kyk
x·d
d
Projd x =
kdk2
cos θ =
−b ±
√
b2 − 4ac
2a